# Construction of abelian varieties from Hilbert modular forms?

Some experts tell me that the construction of abelian varieties from Hilbert modular forms is an (apparently difficult) open problem. However, in view of the construction of $l$-adic Galois representations due to Carayol for instance, it is not clear what exactly the obstruction to the usual method of taking the quotient of the jacobian (of the associated quaternionic Shimura curve) by the `annihilator' of the associated quaternionic eigenform would be.

To be slightly more precise, consider the setting of Carayol "Sur les représentations galoisiennes modulo $l$ attachées aux formes modulaires" (Duke Math. Journal, 1986). That is, let $F$ be a totally real field of degree $d$, with set of real places $\lbrace \tau_1, \ldots, \tau_d \rbrace$. Fix integers $k \geq 2$ and $w$ having the same parity. Let $D_{k,w}$ denote the representation of $\operatorname{GL_2}({\bf{R}})$ that occurs via unitary induction as $\operatorname{Ind}(\mu, \nu)$, where $\mu$ and $\nu$ are the characters on ${\bf{R}}^{\times}$ given by

\begin{align*} \mu(t) &= \vert t \vert ^{\frac{1}{2}(k-1-w)}\operatorname{sgn}(t)^k; ~~ \nu(t) = \vert t \vert ^{\frac{1}{2}(-k+1-w)}. \end{align*} Fix integers $k_1, \ldots k_d$ all having the same parity. Let $\pi \cong \bigotimes_v \pi_v$ be a cuspidal automorphic representation of $\operatorname{GL_2}({\bf{A}}_F)$ such that for each real place $\tau_i$ of $F$, there is an isomorphism $$\pi_{\tau_i} \cong D_{k_i, w}.$$ It is well know that such representations correspond to holomorphic Hilbert modular forms of weight ${\bf{k}}=(k_1, \ldots, k_d)$. If $d$ is even, then assume additionally that there exists a finite prime $v \subset \mathcal{O}_F$ where the local component $\pi_v$ is an "essentially square integrable" (i.e. special or cuspidal) representation of $\operatorname{GL_2}(F_v)$. Let $B/F$ be a quaternion algebra that is ramified at $\lbrace \tau_2, \ldots, \tau_d \rbrace$ if $d$ is odd, and ramified at $\lbrace \tau_2, \ldots, \tau_d, v \rbrace$ if $d$ is even. Let $$G = \operatorname{Res}_{F/{\bf{Q}}}(B^{\times})$$ be the associated algebraic group over ${\bf{Q}}$. Hence, we have an isomorphism $$G({\bf{R}}) \cong \operatorname{GL}({\bf{R}}) \times \left( \mathbb{H}^{\times} \right)^{d-1},$$ where $\mathbb{H}$ denotes the Hamiltonian quaternions. Let $\overline{D}_{k,w}$ denote the representation of $\mathbb{H}^{\times}$ corresponding to $D_{k,w}$ via Jacquet-Langlands correspondence. We then consider cuspidal automorphic representations $\pi' = \bigotimes_v \pi_v'$ of $G({\bf{A}}_F)$ such that $\pi_{\tau_1}' \cong D_{k_1, w}$ and $\pi_{\tau_i} \cong \overline{D}_{k_i, w}$ for $i = 2, \ldots, d$. Such representations should (I believe) correspond to modular forms of weight ${\bf{k}} = (k_1, \ldots, k_d)$ on the indefinite quaternion algebra $B$. To be slightly more precise, let $S_{\bf{k}}(\mathfrak{m})$ denote the finite dimensional ${\bf{C}}$-vector space of quaternionic modular forms of weight ${\bf{k}}$ and level $\mathfrak{m} \subset \mathcal{O}_F$ on $B$. Write $\mathfrak{d} =\operatorname{disc}(B)$. The space $S_{\bf{k}}(\mathfrak{m})$ comes equipped with actions of the standard Hecke operators $T_v$ for all primes $v \nmid \mathfrak{m}\mathfrak{d}$, and with Atkin-Lehner involutions for all prime powers $v^e \mid \mathfrak{m}\mathfrak{d}$. The Jacquet-Langlands correspondence induces a "Hecke equivariant" isomorphism of spaces \begin{align*} S^B_{\bf{k}}(\mathfrak{m}) &\cong S_{\bf{k}}(\mathfrak{m}\mathfrak{d})^{\operatorname{\mathfrak{d}-new}}, \end{align*} where $S_{\bf{k}}(\mathfrak{m}\mathfrak{d})^{\operatorname{\mathfrak{d}-new}}$ denotes the space of cuspidal Hilbert modular forms of weight ${\bf{k}}$ that are new at primes dividing $\mathfrak{d}$.

Anyhow, at least when we assume ${\bf{k}} = (2, \ldots, 2)$, a standard argument shows that there is a $G({\bf{A}}_f)$-equivariant isomorphism $\Gamma(\omega) \cong S^B_{\bf{k}}(\mathfrak{m})$, where $\omega$ is the sheaf of homomorphic $1$-forms on the complex Shimura curve \begin{align*} M(\bf{C}) &= G({\bf{Q}}) \backslash G({\bf{A}}_f) \times X/H.\end{align*} Here, $X = {\bf{C}} - {\bf{R}}$, and $H \subset G({\bf{A}}_f)$ is a compact open subgroup of level $\mathfrak{m}$. Let $M$ denote Shimura's canonical model of this curve (defined over $F$). Let $J$ denote the Jacobian of $M$. Let ${\bf{T}}$ denote the subalgebra of $\operatorname{End}(J)$ generated by Hecke correspondences on $M$. My question is whether or not the following construction can or has been made rigorous. Namely, in the setup above, start with a Hilbert modular eigenform ${\bf{f}} \in \pi$, and consider an associated quaternionic eigenform $\Phi \in \pi'$. Viewing $\Phi$ as an eigenform for the Hecke algebra ${\bf{T}}$, consider the homomorphism $\theta_{\Phi}:{\bf{T}} \longrightarrow E$ that sends a Hecke operator acting on $\Phi$ to its corresponding eigenvalue. Here, $E = E_{\Phi}$ denotes the extension of ${\bf{Q}}$ generated by all of the eigenvalues of $\Phi$. Let $I_{\Phi} = \ker{\theta_{\Phi}}$. Consider the quotient \begin{align*} A &= J/I_{\Phi}J. \end{align*} Is $A$ not an abelian variety associated to the Hilbert modular eigenform ${\bf{f}}$? Or is this completely trivial, with the subtle part being the task of showing that $\dim(A) = [E: {\bf{Q}}]$?

A more naive question to ask is why Shimura's construction cannot be generalized directly for a cuspidal Hilbert modular form ${\bf{f}} \in S_{\bf{k}}(\mathfrak{m})$. Also, how does taking weight ${\bf{k}} = {\bf{2}}$ make the problem simpler? Apologies if parts of this question were somewhat vague, I have sketched matters for simplicity/space.

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There is no problem with constructing an abelian variety $A$ for most Hilbert modular forms of parallel weight $2$, the issue is finding such a variety for all $\pi$. In particular, when $d = [K:\mathbf{Q}]$ is even, there is a local obstruction to the existence of a corresponding Shimura curve which realizes the Galois representation associated to $\pi$. In particular, if $\pi$ has "level one", then no such Shimura curve exists. To construct the Galois representation in this case one has to use congruences; this was done by Taylor in the late 80's. This issue is also discussed here:

Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?

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Thanks for this reply, this is exactly the kind of answer I was looking for! Incidentially, on the topic of conditions imposed by Shimura curves, does the analogous construction via Hilbert modular varieties not work too? That is, start with a Hilbert modular eigenform of parallel weight 2, viewed as a function on some associated Hilbert modular variety Y which equipped with an algebra of Hecke correspondences. Let J be the jacobian of Y, with T the subalgebra of $\End(J)$ generated by Hecke correspondences. The eigenform gives rise to a homomorphism as above, with kernel $I$. Let $A=J/I$? – jvo Feb 7 '11 at 8:32
@vo: Hilbert modular varieties don't have Jacobians, they have Albaneses, and these tend to be trivial! – David Hansen Feb 7 '11 at 22:11
@Barry Lyndon: Oh, I see. I had been wondering why no one seems to work directly with Hilbert modular varieties in these constructions ... I suppose this also explains the importance of Jacquet-Langlands correspondence (to reduce to the study of Shimura curves) in the work of Carayol and others. Thanks for pointing this out. – jvo Feb 8 '11 at 8:33
@David Hansen: Thanks for pointing this out! – jvo Feb 8 '11 at 8:33